You want a repair shop to repair to component level a laptop
motherboard?
Compaq doesn't even fix them to component level and does not have
schematics or individual parts available. The motherboard is the
replacement part.

Since motherboard manufactures do not support having them repaired to
component level, no wonder you are having a hard time finding anyone
willing to even touch it.

You best option is to try and find one on ebay that works but has a
broken display, second option is to try and complain about the Compaq
quality and try and get them to repair it for free as it has been
repaired once for a motherboard failure. Obviously the written
warranty does not apply but the implied warranty does if it is in fact
a repeat failure of the same part, the motherboard.